# A Characterization of Families of Graphs in Which Election Is Possible

## Abstract

**The Model.** We consider networks of processors with arbitrary topology. A network is represented as a connected, undirected graph where vertices denote processors and edges denote direct communication links. Labels (or states) are attached to vertices and edges. Labels are modified locally, that is, on a subgraph of fixed radius 1 of the given graph, according to certain rules depending on the subgraph only (*local computations*). The relabelling is performed until no more transformation is possible, *i.e.*, until a normal form is obtained.

**The Problem.** The election problem is one of the paradigms of the theory of distributed computing. Considering a network of processors the election problem is to arrive at a configuration where exactly one process is in the state *elected* and all other processes are in the state *non-elected* see [Tel00]. The elected vertex is used to make decisions, to centralize or to broadcast some information.

**Known Results.** Graphs where election is possible were already studied but the algorithms usually involved some particular knowledge. Solving the problem for different knowledge has been investigated for some particular cases including (see [Tel00] for details): - the network is known to be a tree - the network is known to be complete - the network is known to be a grid - the nodes have different identification numbers - the network is known to be a ring and has a known prime number of vertices.

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